Vilhelm Hammershoi's show at the Royal Academy proves he should be better known

You may well not have heard of him, but when you come across his work, you'll wonder why you haven't.

An empty room. An open door, giving a glimpse of another empty room. Bare walls. Bare tables. Bare polished boards. A cold, clear light coming in, casting a slant of brightness on to a wall or floor. A woman in a long dark dress, standing alone and motionless. Her back turned. Whites, greys, blacks, browns so deep they're near-monochrome. A geometry of straight edges. The odd gleam. Stillness. Quiet. Mystery. Sorrow. Scenes from a memory.

This is the art of Vilhelm Hammershoi. He was a Danish painter who lived 1864-1916, and specialised in such minimal and highly atmospheric interiors. You may well not have heard of him, but when you come across his work, you'll wonder why you haven't. I'm not quite sure why.

Hammershoi has been waiting quietly in the wings of popularity for some time now. There ought to becalendars and coffee-table books and posters and mouse-mats, but there aren't. It's like he was an acquired taste, but actually he's a taste we already have; we just haven't seen the pictures much. Perhaps the show of his work now on at the Royal Academy of Arts will finally do the trick. To help things along, they've issued a calendar, too.

It was a shortish and single-minded career. Hammershoi seems to have found his signature at once. At 24, he painted a half-length figure. She is standing, face averted, body stationary, looking down at what her hands are doing (which we can't see), against a wide blank wall that pulses with small, soft variations of luminosity.

He never looked back. His art is a succession of rooms, doors, walls, windows, shades of grey and silent women (usually based on his wife, Ida), sitting, standing, reading, doing nothing, revealing nothing. It's cut short only by his death in middle-age from throat cancer. By then he had become quite well known. But after death, his European fame faded.

We critics like a rediscovery, a new artist, a new old artist – someone we can introduce, who brings with him no baggage of clichés and stock responses. But Hammershoi, I must admit, doesn't give the critic much to do, in the way of enthusing or explaining. His emotional impact is so plain, his appeal so immediate.

A marriage of Vermeer and Magritte? Something like that. His work is rich in resonances, of artists both before him and after. Early on, he studied the Dutch interiors of Vermeer and De Hooch. He went to Paris, and presumably saw Chardin's domestic scenes. Did he know Caspar David Friedrich's painting, and its highly charged back-turned figures?

Then there are the echoes of Gwen John, and Magritte, and Edward Hopper, with their modern visions of stasis and solitude and estrangement. They probably didn't know Hammershoi. But the evident likenesses put him at the head of this 20th-century line of minimal melancholy.

His work seems already familiar, a classic somehow forgotten. It's as if he'd extracted an effect that's latent in all those old and modern masters, and distilled it into a pure state. His scenes feel like something remembered. They have the form of a recollection – not a plenitude of detail, but a few distinct and nameable things.

Hammershoi understands the power of negatives, the unsaid, the unshown. His art is full of refusals. The back-turned woman, revealing nothing of her mind; the unmotivated presence of these figures, haunting the rooms like ghosts; the doors closed or ajar, implying something hidden beyond; the apartments' unexplained emptiness, occupied only by light, or with a few isolated signs of life, like the Mary Celeste.

Everything conspires to create a mood of absence, loss, denial. Somebody is dead. Somebody is abandoned. Somebody is fatally repressed. Life is on hold, proceeding in a reduced, trance-like manner. Behind the scenes, on everyone's mind, there's a secret.

And Hammershoi embodies the unending and the unknown with his walls – those blank, flickering expanses, stretches of nothing that face out, flat-on to the picture surface. The paintings' viewers, like the figures in them, do a lot of staring at walls.

It is an intensely narrow art. You might think of Jane Austen's phrase – "the little bit (two Inches wide) of Ivory on which I work with so fine a Brush" – but compared with Hammershoi's, her world is a herd of elephants. And it's not just within the pictures that so little happens, but between them.

Hammershoi's output isn't wholly uniform. Some interiors are obviously more powerful than others, and there are surprising variations on the basic theme. Interior of a Courtyard looks out into a rectangular atrium space within an apartment block, sheer walls of blank closed windows, but with one open (very Hopper). Interior of the Great Hall in Lindgarden reveals a large, ornate, but of course entirely vacant and monochrome, public room, like a ballroom out of season – a big echoing space, with nothing to echo in it.

There are also some minimal and wistful landscapes and townscapes. Yet the overall impression is indeed of little variation. Image after image, room after room – it's like looking at a stop-action film sequence. Figures come and go, in and out, furniture switches place, viewing angles shift, in a scene that remains the same.

An obsessional repetitive theme can be fine. It's the way a lot of modern art carries on, and it's what makes Hammershoi look so modern. But – and it is a rather a big but – it's not the sameyness across his career that's the problem. It's the sameyness within his vision, its sheer straightforwardness. Hammershoi may conjure up mystery, but his art itself holds none.

The RA's catalogue has a fine praising quotation from his contemporary, the poet Rainer Maria Rilke: "Hammershoi is not one of those about whom one must speak quickly. His work is long and slow, and at whichever moment one apprehends it, it will speak of what is important and essential in art."

Like many things Rilke said, this is thoroughly misleading, or misled. It mistakes the kind of experience Hammershoi captures, for the experience his painting offers. Yes, it's a long, slow life we see in these entranced interiors. But the work is not long and slow. It gets to its point, and gets it across, straight away. When Rilke talks about "what is essential in art", that is certainly Hammershoi's pitch. We're meant to see an art reduced to its essentials. But what we actually see is an art reduced to a consistency.

No art that's really worth looking at is simple. It may seem simple, but to hold us, there has to be some grit or subtext or counter-current or internal tension. And there isn't in Hammershoi's painting. It's a very original and distinctive art. It's remarkably effective in its mood-making, precisely because all its elements converge to a single point. But its lack of complication makes it thin and flat. The mood, as it strikes again and again, loses its magic. You realise you're in the presence of a formula, delivered pat. Hammershoi's purism is a kind of professionalism.

The RA likes to offer visitors appropriate reading matter. For this exhibition, the works of the Danish philosopher Kierkegaard are on sale. I'm not sure if that severe and hysterical character is a particularly close match for Hammershoi, but something that Kierkegaard said strikes a chord: "Purity of heart is to will one thing." Change heart to art, and it might be Hammershoi's motto. Don't you believe it, though. The purest art always has to have another thing, too.

Vilhelm Hammershoi – The Poetry of Silence, Royal Academy of Arts, London W1 (020-7300 8000), to 7 September